Impacts of the cross-correlated noises on the fluctuation behaviors of a gene transcriptional regulatory system

The fluctuation behaviors of a gene transcriptional regulatory system subjected to the cross-correlated noises are investigated. Based on the approximate Fokker-Planck Equation, the analytic expressions of the associate relaxation time and the correlation function of a gene transcriptional regulatory system are obtained by means of projection operator method. The results show that the noise parameters exhibit different effects on the fluctuation behaviors of this system, and can produce the critical behaviors in the relaxation time as functions of the multiplicative noise intensity D, the additive noise intensity α and the cross-correlated noise intensity λ. That is, the relaxation time decreases below the critical value of λ, and above the critical value of D and α. The rate of fluctuation decay of concentration state of TF-A is speeded up, the stability of this system is enhanced. However, above the critical value of λ, and below the critical values of D and α, the relaxation time first increases, reaches a maximum, and then decreases. The rate of fluctuation decay of concentration state of TF-A is slowed down firstly, and then, is enhanced. On the other hand, the effects of noise parameters on the correlation function show that λ and α weaken the related activity between two states at different time, and enhances the stability of the system in the steady state. In contrast, D enhances the related activity between two states at different time, and weakens the stability. The numerical results are in basic agreement with the theoretical predictions.